A098646 -id:A098646 - cp's OEIS frontend

A098647 Trace sequence associated to the 4 X 4 Krawtchouk matrix and its transpose.

1, 12, 224, 4608, 96256, 2015232, 42205184, 883949568, 18513657856, 387755016192, 8121246285824, 170093589823488, 3562486393470976, 74613683694600192, 1562729279488262144, 32730226951263879168

Offset: 0

Paul Barry, Sep 18 2004

Let A=[1,1,1,1;3,1,-1,-3;3,-1,-1,3;1,-1,1,-1], the 4 X 4 Krawtchouk matrix. Then a(n)=trace((A*A')^n)/4.

Twelfth binomial transform of ((4*sqrt(5))^n +(-4*sqrt(5))^n)/2, with g.f. 1/(1-80*x^2).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (24,-64).

Cf. A098646.

G.f.: (1-12*x)/(1-24*x+64*x^2).
a(n) = ((12+4*sqrt(5))^n+(12-4*sqrt(5))^n)/2.
a(n) = 2^(n-1)*((sqrt(5)-1)^(2*n)+(sqrt(5)+1)^(2*n)).
a(n) = 4^n*A098648(n). - R. J. Mathar, Nov 11 2013

A282124 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 430", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 3, 15, 11, 63, 43, 255, 171, 1023, 683, 4095, 2731, 16383, 10923, 65535, 43691, 262143, 174763, 1048575, 699051, 4194303, 2796203, 16777215, 11184811, 67108863, 44739243, 268435455, 178956971, 1073741823, 715827883, 4294967295, 2863311531, 17179869183

Offset: 0

Robert Price, Feb 06 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A282121, A282122, A282123.

Cf. A001045, A010684, A024036, A081631, A098646.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 430; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]

Conjectures from Colin Barker, Feb 07 2017: (Start)
a(n) = (-1 + 2*(-1)^n - (-1)^n*2^(1+n) + 2^(2+n)) / 3.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
(End)
Conjectures from Paul Curtz, Jun 10 2019: (Start)
a(n) = A001045(n+1)*(period 2: repeat[1, 3]).
a(n+4) = a(n) + 10*A081631(n).
a(2*n+1) = 2^(2*n+2) -1.
a(n+2) = a(n) + A098646(n+1).
(End)

A308663 Partial sums of A097805.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 16, 17, 21, 27, 31, 32, 32, 33, 38, 48, 58, 63, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 129, 136, 157, 192, 227, 248, 255, 256, 256, 257, 265, 293, 349, 419, 475, 503, 511, 512

Offset: 0

Paul Curtz, Jun 15 2019

Curtz (1965), page 15, from right to left, gives (F1):
  1/2;
  1/4,  3/4;
  1/8,  4/8,   7/8;
  1/16, 5/16, 11/16, 15/16;
  ... .
Numerators + Denominators = (C) =
   3;
   5,  7;
   9, 12, 15;
  17, 21, 27, 31;
  ... .
This is the current sequence without powers of 2.
The triangle (P) for a(n) is
  1;
  1, 2;
  2, 3,  4;
  4, 5,  7,  8;
  8, 9, 12, 15, 16;
  ... .
(C) is the core of (P).
Extension of (F1). (F2) =
  0/1;
  0/1, 1/1;
  0/2, 1/2, 2/2;
  0/4, 1/4, 3/4, 4/4;
  0/8, 1/8, 4/8, 7/8, 8/8;
  ... .
(Mentioned, without 0's, op. cit., page 16.)
a(n) = Numerators + Denominators.
Row sums of triangle (P): A084858(n).
From right to left, with alternating signs: 1, 1, 3, 2, 12, 8, 48, 32, ..., see A098646.
For triangle (C), row sums give A167667(n+1).
From right to left, with alternating signs: A098646(n).
Rank of A016116(n): 0 together with A117142.

Paul Curtz, Accélération de la convergence de certaines séries alternées à l'aide des fonctions de sommation, Thèse de 3ème Cycle d'Analyse Numérique, Faculté des Sciences de l'Université de Paris, 4 mai 1965.

Cf. A097805.

T(n,k) = ceiling(2^(n-1)) + Sum_{j=0..k-1} binomial(n-1,j). - Alois P. Heinz, Jun 15 2019

a(n+1) = a(n) + A097805(n+1) for n >= 0.

Edited by N. J. A. Sloane, Sep 15 2019

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A098647 Trace sequence associated to the 4 X 4 Krawtchouk matrix and its transpose.

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A282124 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 430", based on the 5-celled von Neumann neighborhood.

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A308663 Partial sums of A097805.

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